An accuracy analysis of Galerkin meshfree methods accounting for numerical integration

Abstract

Presently, it has been widely recognized that numerical integration plays a crucial role on the solution accuracy of Galerkin meshfree methods due to the rational nature of meshfree shape functions and the misalignment of shape function supports with integration cells. Nonetheless, the corresponding underlying cause is still not quite clear. To address this issue, an accuracy analysis of Galerkin meshfree methods is presented in this study with a special focus on the contribution of numerical integration to the error estimates. One important fact is that the integration difficulty of meshfree methods leads to a loss of the Galerkin orthogonality condition which usually serves as a basis to develop error bounds. In order to enable the accuracy analysis, an error measure is proposed to evaluate the loss of Galerkin orthogonality condition, which is actually related to the order of integration consistency for Galerkin meshfree formulation. With the aid of this orthogonality error measure, both H1 and L2 error estimates are established for Galerkin meshfree methods. It turns out that these errors essentially consist of two parts, one comes from the standard interpolation error, and the other one attributes to the numerical integration error. In accordance with the proposed error estimates, it is evident that for the conventional Gauss integration schemes violating the integration consistency, the solution errors eventually are controlled by the integration error and optimal convergence cannot be achieved even with high order quadrature rules. On the other hand, for the consistent integration approaches, e.g., the stabilized conforming nodal integration and the more recent reproducing kernel gradient smoothing integration, the integration and interpolation errors have the same accuracy order and thus an optimal convergence of Galerkin meshfree methods is ensured. The proposed theoretical error estimates for Galerkin meshfree methods are well demonstrated by numerical results.